9t^2-72t-54=0

Solution for 9t^2-72t-54=0 equation:

Simplifying
9t2 + -72t + -54 = 0

Reorder the terms:
-54 + -72t + 9t2 = 0

Solving
-54 + -72t + 9t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '9'.
9(-6 + -8t + t2) = 0

Ignore the factor 9.
Subproblem 1
Set the factor '(-6 + -8t + t2)' equal to zero and attempt to solve:

Simplifying
-6 + -8t + t2 = 0

Solving
-6 + -8t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -8t + 6 + t2 = 0 + 6

Reorder the terms:
-6 + 6 + -8t + t2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -8t + t2 = 0 + 6
-8t + t2 = 0 + 6

Combine like terms: 0 + 6 = 6
-8t + t2 = 6

The t term is -8t.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8t + 16 + t2 = 6 + 16

Reorder the terms:
16 + -8t + t2 = 6 + 16

Combine like terms: 6 + 16 = 22
16 + -8t + t2 = 22

Factor a perfect square on the left side:
(t + -4)(t + -4) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(t + -4) equal to 4.69041576 and -4.69041576.

Subproblem 1
t + -4 = 4.69041576

Simplifying
t + -4 = 4.69041576

Reorder the terms:
-4 + t = 4.69041576

Solving
-4 + t = 4.69041576

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '4' to each side of the equation.
-4 + 4 + t = 4.69041576 + 4

Combine like terms: -4 + 4 = 0
0 + t = 4.69041576 + 4
t = 4.69041576 + 4

Combine like terms: 4.69041576 + 4 = 8.69041576
t = 8.69041576

Simplifying
t = 8.69041576

Subproblem 2
t + -4 = -4.69041576

Simplifying
t + -4 = -4.69041576

Reorder the terms:
-4 + t = -4.69041576

Solving
-4 + t = -4.69041576

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '4' to each side of the equation.
-4 + 4 + t = -4.69041576 + 4

Combine like terms: -4 + 4 = 0
0 + t = -4.69041576 + 4
t = -4.69041576 + 4

Combine like terms: -4.69041576 + 4 = -0.69041576
t = -0.69041576

Simplifying
t = -0.69041576

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {8.69041576, -0.69041576}
Solution
t = {8.69041576, -0.69041576}